The Bézout equation on the right half-plane in a Wiener space setting

Abstract

This paper deals with the Bézout equation G(s)X(s)=Im,Rs≤0, in the Wiener space of analytic matrix-valued functions on the right halfplane. In particular, G is an m × p matrix-valued analytic Wiener function, where p ≥ m, and the solution X is required to be an analyticWiener function of size p × m. The set of all solutions is described explicitly in terms of a p × p matrix-valued analyticWiener function Y , which has an inverse in the analytic Wiener space, and an associated inner function Θ defined by Y and the value of G at infinity. Among the solutions, one is identified that minimizes the H 2- norm. A Wiener space version of Tolokonnikov's lemma plays an important role in the proofs. The results presented are natural analogs of those obtained for the discrete case in [11]